We now give a characterization of (REF ) for GWF populations, in the same spirit as is done in [1]} for exchangeable offspring distributions.
Our technique, and analogous results, should apply to more general cases.
Let us introduce further notation. We call \((s_1,s_2,\dots )\)  a
size-biased reordering of a random probability vector \(\eta ^{(N)}=\left(\eta ^{(N)}_1,\dots ,\eta ^{(N)}_N\right)\)  if, conditional on \(\eta ^{(N)}\) 
and letting \(K=\sum _{j=1}^N \mathbb {1}_{\eta ^{(N)}_j>0}\)  to be the number of non-zero entries of \(\eta ^{(N)}\) ,
the vector \((s_1,\dots ,s_{K})\)  is a size-biased reordering of the non-zero entries of \(\eta ^{(N)}\)  in the usual sense,
and \(s_j0\)  for \(j>K\) .
